1. Field of the Invention.
The present invention relates generally to signal decomposition methods and apparatus and, more particularly, but not by way of limitation, to systems for providing representations of transient signals.
2. Brief Description of the Prior Art.
When a person measures or records a signal of some type; for example, the output of a microphone or geophone, he is generally not interested in the signal per se. Rather, the signal carries information and it is the information carried by the signal that he is after.
Very often, the recordation or measurement of the signal is only a first step in obtaining the information. From an information point of view, the signal, which expresses the variation of a physical quantity with time, is merely one of various representations of the information the signal contains. For example, it is well known that information expressed as a varying signal in the time domain can also be expressed as a combination of pure tones having different amplitudes, frequencies and phases in the frequency domain. In some cases, a time domain representation of the information is well suited to the information gatherer's purposes and in other cases, a frequency domain representation will be better suited to these purposes.
However, there exist situations in which neither of these representations of information which is obtainable by recording a signal is particularly well suited to the purpose for which the information has been gathered. Such is particularly true when the phenomenon underlying the production of the signal is transient in nature. In such circumstances, it may well be the case that neither a pure time domain representation nor a pure frequency domain representation will provide the information gatherer with the type of data he needs to make use of the information. Because of the inadequacy of these representations of information carried by a signal to express the information in a useable form, a hybrid representation, having both time and frequency aspects, was proposed by Dennis Gabor in 1946. Such proposal has been summarized in Technical Report No. 238 by D. Gabor, published by the Research Laboratory of Electronics at the Mass. Institute of Technology on April 3, 1952.
In the hybrid representation suggested by Gabor, a signal would be decomposed into a plurality of wavelets which would each be associated with a cell in the combined time-frequency domain which Gabor referred to as the information plane. The cells would form a rectangular grid so that each cell would be defined by a fixed interval in each of the time and frequency domains. The wavelets proposed by Gabor had the general form of a sinusoid contained within a gaussian envelope and, in order that the representation, once obtained, would contain the maximum information theoretically possible, each cell had unit area. An important aspect of the wavelet proposed by Gabor, as well as wavelets with which the present invention is concerned, is that the wavelets have both time and frequency characteristics. That is, the cell with which the wavelet is associated is defined by a particular frequency interval and a particular time extent. This characteristic makes the wavelets particularly well suited for representing transient signals.
While the hybrid representation proposed by Gabor held great promise for extracting information contained in a transient signal, the proposal was subject to practical difficulties which have resulted in the proposal not having been implemented prior to the present invention. In particular, Gabor recognized the non-orthogonality of his representation, but, indeed, the main difficulty was due to the fact that for the wavelet described by Gabor the higher the frequency, the higher the number of periods in the wavelet. It follows that the Gabor expansion presents the same shortcomings as the Fourier expansion for high frequencies; that is, undersampling. As a result, Gabor's suggestion lay dormant until about 1980 when further theoretical developments were made by Jean Morlet, such developments having been published in a series of articles beginning in 1981.
To overcome the difficulties associated with the time-frequency representation proposed by Gabor, Morlet proposed that a non-rectangular grid be used to effect the decomposition. In particular, Morlet suggested that the frequency scale be divided into suboctave bands, rather than constant frequency intervals, and that a different time scale be used for each band. In particular, the time scale for each band would be chosen to be a fixed portion of a standard wavelet, expressed in cycles, defined by limiting a sinusoid having the frequency associated with the band by an envelope that has appreciable values for only a few periods of the sinusoid. That is, the general form of the wavelet would be the same for all cells so that the wavelet can be generally defined and, after such definition, used to establish time scales for the various bands. It will be noted that a wavelet defined for one cell would differ from a wavelet defined for another cell only by a translation and a dilation or contraction in the time domain. Morlet was then able to show that the cells so defined would have equal areas in the time-frequency domain so that such a representation would be capable of at least approaching the theoretical maximum information contained within the signal so represented, and more particularly would give a good representation of the phase information.
However, Morlet's proposal introduced a new problem with respect to a practical implementation of the time-frequency representation of a signal. To obtain the maximum information available in a signal, the bands and wavelet portions used to define the grid in the time-frequency plane give rise to complex relationships between the time extents of the cells. Thus, a mismatch exists between time intervals defined on the cycle-octave grid and real times defined by clocks which would have to be used to specify a practical time-frequency representation of a signal. The present invention is based on a further theoretical development which has solved this grid time-clock time mismatch. Such development will be discussed below.